Good to know there seems to be an effort to keep some consistency.
(The trickiest thing is actually brightness. LG originally used to set brightness to 100 nits in Filmmaker Mode for SDR, which is correct dark room behavior -- but a lot of people aren't in dark rooms and want brighter screens, so they changed it to be significantly brighter. Defensible, but it now means that if you are in a dark room, you have to look up which brightness level is close to 100 nits.)
Game mode will indeed likely turn off any expensive latency-introducing processing but it's unlikely to provide the best color accuracy.
It makes me wish that there was something like an industry standard 'calibrated' mode that everyone could target - let all the other garbage features be a divergence from that. Hell, there probably is, but they'd never suggest a consumer use that and not all of their value-add tackey DSP.
“Filmmaker Mode” on LG OLED was horrible. Yes, all of the “extra” features were off, but it was overly warm and unbalanced as hell. I either don’t understand “Filmmakers” or that mode is intended to be so bad that you will need to fix it yourself.
TV producers always set their sets to way higher by default because blue tones show off colors better.
As a result of both that familiarity and the better saturation, most people don't like filmmaker when they try to use it at first. After a few weeks, though, you'll be wondering why you ever liked the oversaturated neons and severely off brightness curve of other modes.
Or not, do whatever you want, it's your TV!
When you say that "HDR is static" you probably mean that "Dynamic tone-mapping" was turned off. This is also correct behavior. Dynamic tone-mapping isn't about using content settings to do per-scene tone-mapping (that's HDR10+ or Dolby Vision, though Samsung doesn't support the latter), it's about just yoloing the image to be brighter and more vivid than it should be rather than sticking to the accurate rendering.
What you're discovering here is that the reason TV makers put these "garbage features" in is that a lot of people like a TV picture that's too vivid, too blue, too bright. If you set it to the true standard settings, people's first impression is that it looks bad, as yours was. (But if you live with it for a while, it'll quickly start to look good, and then when you look at a blown-out picture, it'll look gross.)
I'd suggest living with it for a while; if you do, you'll quickly get used to it, and then going to the "standard" (sic) setting will look too blue.
What is the correct colour?
Well for starters you’re viewing the real sky in 3D and your TV is a 2D medium. Truly that immediately changes your perception and drastically. TV looks like TV no matter what.
It is not as clear cut as you think and is very much a gradient. I could send 10 different color gradings of the sky and grass to 10 different people and they could all say it looks “natural” to them, or a few would say it looks “off,” because our expectations of “natural” looks are not informed by any sort of objective rubric. Naturally if everyone says it’s off the common denominator is likely the colorist, but aside from that, the above generally holds. It’s why color grading with proper scopes and such is so important. You’re doing your best to meet the expectation for as many people as possible knowing that they will be looking on different devices, have different ideas of what a proper color is, are in different environments, etc. and ultimately you will still disappoint some folks. There are so many hardware factors at play stacked on top of an individual’s own expectations.
Even the color of the room you’re in or the color/intensity of the light in your peripheral vision will heavily influence how you perceive a color that is directly in front of you. Even if you walk around with a proper color reference chart checking everything it’s just always going to have a subjective element because you have your own opinion of what constitutes green grass.
There is no such thing as the “correct” or “most natural” image. There is essentially no “true” image.